Device and method for capacitive sensing

ABSTRACT

A semiconductor device comprises driver circuitry, an analog-digital (AD) converter, and processing circuitry. The driver circuitry is configured to supply a drive signal to a sensor array in a sensing frame comprising 2N bursts, N being an integer of two or more. The mixer circuitry is configured to modulate a plurality of carrier waves with a plurality of sensing signals corresponding to capacitances of a plurality of sensing electrodes of the sensor array, respectively, to output a plurality of mixer outputs. A number of the plurality of sensing electrodes is 2N−1 or 2N. The AD converter is configured to perform AD conversion on a sum signal of the plurality of mixer outputs. The processing circuitry is configured to detect an object based on the output of the AD converter.

CROSS REFERENCE

This application is a continuation of U.S. patent application Ser. No. 17/062,529 filed Oct. 2, 2020, which is a continuation of U.S. patent application Ser. No. 16/533,490 filed Aug. 6, 2019, now U.S. Pat. No. 10,795,503, which claims priority to Japanese Patent Application No. 2018-150310, filed on Aug. 9, 2018. The disclosure of the aforementioned patent applications are incorporated herein by reference in their entireties.

BACKGROUND Field

The present disclosure relates to a device and method for capacitive sensing.

Description of the Related Art

Capacitive sensing is used for proximity sensing to detect an approach or contact of an object, e.g., a user's finger or a stylus, to an input device, e.g., a touch panel. While performing capacitive sensing, motion artifacts may be detected. As the motion artifacts may generate errors while performing capacitive sensing, a capacitive sensing system may be designed to suppress detection of a motion artifact.

SUMMARY

In one or more embodiments, a semiconductor device comprises driver circuitry, mixer circuitry, an analog-digital (AD) converter, and a processing circuitry. The driver circuitry is configured to supply a drive signal to a sensor array in a sensing frame comprising 2N bursts, N being an integer of two or more. The mixer circuitry is configured to modulate a plurality of carrier waves with a plurality of sensing signals corresponding to capacitances of a plurality of sensing electrodes of the sensor array, respectively, to output a plurality of mixer outputs. The number of the plurality of sensing electrodes being 2N−1 or 2N. The AD converter is configured to perform AD conversion on a sum signal of the plurality of mixer outputs. The processing circuitry is configured to perform proximity sensing of an object to the sensor array with respect to the sensing frame based on the output of the AD converter. A carrier wave modulated with one of the plurality of sensing signals is selected from first and second carrier waves during a plurality of first bursts of the 2N bursts. The first carrier wave being in phase with the drive signal, and the second carrier wave has a phase shifted from that of the first carrier wave by π.

In one or more embodiments, a method comprises supplying a drive signal to a sensor array in a sensing frame comprising 2N bursts. N being an integer of two or more. The method further comprises modulating a plurality of carrier waves with a plurality of sensing signals corresponding to capacitances of a plurality of sensing electrodes of the sensor array, respectively, to output a plurality of mixer outputs. The number of the plurality of sensing electrodes being 2N−1 or 2N. Further, the method comprises performing AD conversion on a sum signal of the plurality of mixer outputs to generate a digital data stream. Additionally, the method comprises performing proximity sensing of an object to the sensor array with respect to the sensing frame based on the digital data stream. A carrier wave modulated with one of the plurality of sensing signals is selected from first and second carrier waves during a plurality of first bursts of the 2N bursts. The first carrier wave being in phase with the drive signal, and the second carrier wave has a phase shifted from that of the first carrier wave by Tr.

BRIEF DESCRIPTION OF THE DRAWINGS

So that the manner in which the above recited features of the present disclosure may be understood in detail, a more particular description of the disclosure, briefly summarized above, may be had by reference to embodiments, some of which are illustrated in the appended drawings. It is to be noted, however, that the appended drawings illustrate only some embodiments of this disclosure and are therefore not to be considered limiting of its scope, for the disclosure may admit to other equally effective embodiments.

FIG. 1 is a block diagram illustrating an example configuration of a capacitive sensing system, according to one or more embodiments.

FIG. 2 is a circuit diagram illustrating an example configuration of capacitive sensing circuitry, according to one or more embodiments.

FIG. 3 illustrates an example operation of the capacitive sensing system, according to one or more embodiments.

FIG. 4 is a circuit diagram illustrating an example configuration of capacitive sensing circuitry, according to one or more embodiments.

FIGS. 5, 6, 7, and 8 are circuit diagrams illustrating an example operation of the capacitive sensing circuitry, according to one or more embodiments.

FIG. 9 is a block diagram illustrating a capacitive sensing system, according to one or more embodiments.

DETAILED DESCRIPTION

A description is given below of embodiments of this disclosure. In the following, same or similar components may be denoted by same or corresponding reference numerals. Suffixes may be attached to reference numerals to distinguish same components from each other.

In one or more embodiments, as illustrated in FIG. 1, a capacitive sensing system 100 comprises a sensor array 1 and a semiconductor device 2.

In one or more embodiments, the sensor array 1 comprises sensing electrodes 3 arrayed in rows and columns. In one or more embodiments, the sensor array 1 may be integrated in a touch panel. In alternative embodiments, the sensor array 1 may be integrated in a display panel, such as a liquid crystal display (LCD) panel and an organic light emitting diode (OLED) display panel. When the sensor array 1 is integrated in an LCD panel comprising a plurality of common electrodes arrayed in rows and column, the common electrodes may be used as the sensing electrodes 3, in one or more embodiments. In one or more embodiments, the sensing electrodes 3 are connected to the semiconductor device 2.

In one or more embodiments, the semiconductor device 2 comprises capacitive sensing circuitry 4 and a processing unit 5. In one or more embodiments, the capacitive sensing circuitry 4 is configured to obtain sensing signals having signal levels corresponding to capacitances of the sensing electrodes 3 and generate digital outputs based on the sensing signals. The processing unit 5 is configured to perform arithmetic processing on the digital outputs received from the capacitive sensing circuitry 4 to achieve proximity sensing of an object, such as a user's finger and a stylus, to the sensor array 1. In one or more embodiments, the arithmetic processing for the proximity sensing is performed in units of Q sensing electrodes arrayed in a specific direction, for example, the vertical direction in FIG. 1, where Q is an integer of three or more. When the sensor array 1 is integrated in a display panel, display driver circuitry configured to drive the display panel may be integrated in the semiconductor device 2.

In one or more embodiments, as illustrated in FIG. 2, the capacitive sensing circuitry 4 comprises input terminals 11 ₁ to 11 _(Q) connected to the sensing electrodes 3 ₁ to 3 _(Q), respectively, driver circuitry 12, current conveyer circuitry 13, encoder circuitry 14, an analog-digital (AD) converter 15, and decoder circuitry 16. Illustrated in FIG. 2 is the configuration of a part of the capacitive sensing circuitry 4 related to the Q sensing electrodes 3 enclosed in the dotted box 6 in FIG. 1. In one or more embodiments, the Q sensing electrodes 3 illustrated in FIG. 2 are arrayed in the vertical direction in FIG. 1. In one or more embodiments, one AD converter 15 is prepared for Q sensing electrodes 3.

In one or more embodiments, the driver circuitry 12 comprises Q drivers 21 ₁ to 21 _(Q) connected to the input terminals 11 ₁ to 11 _(Q), respectively. In one or more embodiments, the drivers 21 ₁ to 21 _(Q) are configured to supply drive signals to the sensing electrodes 3 ₁ to 3 _(Q), respectively. In one or more embodiments, the drive signals are periodic signals of a frequency of f_(REF). In one or more embodiments, the drivers 21 ₁ to 21 _(Q) are supplied with a reference clock signal CLK_(REF) of the frequency f_(REF), and configured to generate the drive signals in synchronization with the reference clock signal CLK_(REF). Signals of various waveforms may be used as the drive signals. For example, signals of a sinusoidal, triangular, or rectangular waveform may be used as the drive signals.

In one or more embodiments, the current conveyer circuitry 13 comprises current conveyers 22 ₁ to 22 _(Q) connected to the input terminals 11 ₁ to 11 _(Q), respectively. In one or more embodiments, the current conveyers 22 ₁ to 22 _(Q) are configured to output sensing signals f₁ to f_(Q) having signal levels corresponding to current levels of charging and discharging currents of the sensing electrodes 3 ₁ to 3 _(Q), respectively. In one or more embodiments, the charging and discharging currents of the sensing electrodes 3 ₁ to 3 _(Q) depend on the self-capacitances of the sensing electrodes 3 ₁ to 3 _(Q) when the drive signals are supplied to the sensing electrodes 3 ₁ to 3 _(Q), and the sensing signals f₁ to f_(Q) are obtained as signals generated by modulating the periodic signals of the frequency f_(REF) with information of the self-capacitances of the sensing electrodes 3 ₁ to 3 _(Q).

In one or more embodiments, the encoder circuitry 14 comprises a phase shifter 23 and Q mixers 24 ₁ to 24 _(Q). In one or more embodiments, the phase shifter 23 is configured to supply carrier waves Mix1 to MixQ to the mixers 24 ₁ to 24 _(Q), respectively, in synchronization with the reference clock signal CLK_(REF). In one or more embodiments, the mixers 24 ₁ to 24 _(Q) are configured to generate mixer outputs g₁ to g_(Q) by modulating the carrier waves Mix1 to MixQ with the sensing signals f₁ to f_(Q), respectively.

In one or more embodiments, the carrier waves Mix1 to MixQ are each selected from signals ϕ_(−π/2), ϕ₀, ϕ_(π/2), and ϕ_(π). In one or more embodiments, the signal ϕ₀ is a signal in phase with the drive signals. In one or more embodiments, the signal ϕ₀ is a periodic signal having the same phase as that of the current waveforms of the drive signals. In one or more embodiments, the signal ϕ_(π) is a signal of the opposite phase to that of the drive signals. In one or more embodiments, the signal ϕ_(π/2) is a periodic signal having a phase delayed from that of the signal ϕ₀ by π/2, and the signal ϕ_(−π/2) is a periodic signal having a phase advanced from that of the signal ϕ₀ by π/2, that is, a phase delayed from that of the signal ϕ₀ by 3π/2.

In one or more embodiments, a sum signal of the mixer outputs g₁ to g_(Q) outputted from the mixers 24 ₁ to 24 _(Q) is supplied to the AD converter 15. In one or more embodiments, the mixer outputs g₁ to g_(Q) are current signals, and the outputs of the mixer 24 ₁ to 24 _(Q) are commonly connected to the input of the AD converter 15 to achieve the addition of the mixer outputs g₁ to g_(Q) through wired-sum. FIG. 2 illustrates such a configuration.

When the carrier wave Mixk, which is modulated with the sensing signal f_(k), is selected from the signals ϕ₀ and ϕ_(π), the mixer output g_(k) outputted from the mixer 24 _(k) comprises information of an effective signal component for proximity sensing, in one or more embodiments. When the carrier wave Mixk is selected from the signals ϕ_(π/2) and ϕ_(−π/2), the mixer output g_(k) comprises information of a noise component, in one or more embodiments. The signal level of the mixer output g_(k) is originally expected to be zero, when the carrier wave Mixk is selected from the signals ϕ_(π/2) and ϕ_(−π/2), in one or more embodiments. In one or more embodiments, when the signal level of the mixer output g_(k) is not zero, this indicates that the sensing signal f_(k) experiences an influence of noise. In one or more embodiments, the mixer output g_(k) may be used for noise detection, when the carrier wave Mixk is selected from the signals ϕ_(π/2) and ϕ_(−π/2).

A periodic signal of a different phase from that of the signals ϕ₀ and ϕ_(π) may be used in place of the signal ϕ_(π/2) or ϕ_(−π/2). In one or more embodiments, the signal level of the mixer output g_(k) incorporates information of a noise component when a periodic signal of a different phase from that of the signals ϕ₀ and ϕ_(π) is used as the carrier wave Mixk. Also in this case, the mixer output g_(k) may be used for noise detection.

In one or more embodiments, the configuration of the encoder circuitry 14 achieves code division multiplexing. In one or more embodiments, codes are assigned to the sensing signals f₁ to f_(Q). In one or more embodiments, the phase shifter 23 is configured to select the carrier wave Mixk from among the signals ϕ_(−π/2), ϕ₀, ϕ_(π/2) and ϕ_(π) based on the code assigned to the sensing signal f_(k) and supply the selected carrier wave Mixk to the mixer 24 _(k).

In one or more embodiments, the AD converter 15 is configured to perform analog-digital conversion on the sum signal of the mixer outputs g₁ to g_(Q) received from the mixers 24 ₁ to 24 _(Q) to output a digital data stream q.

In one or more embodiments, the decoder circuitry 16 constitutes processing circuitry configured to process the output of the AD converter 15, that is, the digital data stream q, together with the processing unit 5. In one or more embodiments, the decoder circuitry 16 is configured to decode the digital data stream q received from the AD converter 15 to generate the digital outputs. In one or more embodiments, the decoder circuitry 16 is configured to perform a product-sum operation to generate the digital outputs. In one or more embodiments, the processing unit 5 is configured to perform arithmetic processing on the digital outputs from the decoder circuitry 16 to achieve proximity sensing of an object to the sensor array 1. In one or more embodiments, this arithmetic processing may comprise processing for detecting the position at which the object approaches or is in contact with the sensor array 1.

In one or more embodiments, Q is an even number of four or more. In this case, Q can be represented as 2N, where N is an integer of two or more. It should be noted that the technologies of this disclosure are applicable to a case where Q is an odd number of three or more, as described later.

In one or more embodiments, as illustrated in FIG. 3, proximity sensing is performed in each sensing frame. In one or more embodiments, each sensing frame comprises Q bursts, that is, 2N bursts. The 2N bursts may be referred to as bursts #1 to #(2N), respectively. In one or more embodiments, the drivers 21 ₁ to 212 _(N) are configured to continue outputting drive signals over bursts #1 to #(2N) with a constant waveform.

In one or more embodiments, when a drive signal is supplied to the sensing electrode 3 i with a constant waveform over bursts #1 to #(2N), the waveform of the sensing signal f₁ is expected to remain unchanged over bursts #1 to #(2N), and therefore the waveform of the mixer output g_(i) outputted from the mixer 24 _(i) is also expected to remain unchanged over bursts #1 to #(2N). It is therefore expected that the following expressions (1) hold:

$\begin{matrix} {{{{A_{1}(t)} = {{A_{1}\left( {t + T} \right)} = {\ldots = {A_{1}\left( {t + {\left( {{2N} - 1} \right)T}} \right)}}}},{{A_{2}(t)} = {{A_{2}\left( {t + T} \right)} = {\ldots = {A_{2}\left( {t + {\left( {{2N} - 1} \right)T}} \right)}}}},\ldots}{{{A_{2N}(t)} = {{A_{2N}\left( {t + T} \right)} = {\ldots = {A_{2N}\left( {t + {\left( {{2N} - 1} \right)T}} \right)}}}},}} & (1) \end{matrix}$

where t, t+T, . . . and t+(2N−1)T are the times when the AD converter 15 samples the input signal during burst #1 to #(2N), respectively, and A₁(t) to A_(2N)(t) are the signal levels of the sensing signals f₁ to f_(2N) at time t.

For simplicity, the signal levels of the sensing signals f₁ to f_(2N) at times t, t+T . . . and t+(2N−1) may be simply referred to as A₁ to A_(2N). The signal levels on the inputs of the encoder circuitry 14, that is, the signal levels of the sensing signals f₁ to f_(2N) may be collectively represented by an input tensor a, which is represented by the following expression (2):

$\begin{matrix} {a = {\begin{pmatrix} A_{1} \\ A_{2} \\ \vdots \\ A_{2N} \end{pmatrix}.}} & (2) \end{matrix}$

In one or more embodiments, the processing performed by the encoder circuitry 14 during bursts #1 to #(2N) can be expressed as an encode tensor E defined by the following expression (3):

$\begin{matrix} {{E = \begin{pmatrix} e_{11} & \ldots & e_{1{({2N})}} \\ \vdots & \ddots & \vdots \\ e_{{({2N})}1} & \ldots & e_{{({2N})}{({2N})}} \end{pmatrix}},} & (3) \end{matrix}$

where the j^(th) row of the encode tensor E represents the processing performed during burst #j by the encoder circuitry 14, and the k^(th) column represents the modulation using the sensing signal f_(k). For a case where the mixer 24 ₁ to 24 _(Q) all have a gain of 1, for example, the element e_(jk) of the encode tensor E is e^(i0) (=1), e^(iπ/2) (=i), e^(iπ)(=−1), and e^(−iπ/2) (=−i) when the carrier wave Mixk, which is modulated with the sensing signal f_(k), is the signal ϕ₀, ϕ_(π/2), ϕ_(π), and ϕ_(−π/2), respectively, during burst #j.

In one or more embodiments, the digital data stream q outputted from the AD converter 15 can be represented by the following expression (4):

$\begin{matrix} {{\begin{pmatrix} q_{1} \\ q_{2} \\ \vdots \\ q_{2N} \end{pmatrix} = {{Ea} = {\begin{pmatrix} e_{11} & \ldots & e_{1{({2N})}} \\ \vdots & \ddots & \vdots \\ e_{{({2N})}1} & \ldots & e_{{({2N})}{({2N})}} \end{pmatrix}\begin{pmatrix} A_{1} \\ A_{2} \\ \vdots \\ A_{2N} \end{pmatrix}}}},} & (4) \end{matrix}$

-   -   where q₁ to q_(2N) are the values of the digital data stream q         during bursts #1 to #(2N), respectively.

In one or more embodiments, when the carrier wave Mixk, which is modulated with the sensing signal f_(k), is selected from the signals ϕ₀ and ϕ_(π) during a certain burst, the sum signal supplied to the AD converter 15 and the digital data stream q outputted from the AD converter 15 comprise information of a signal component of the sensing signal f_(k) effective for proximity sensing during the certain burst. In one or more embodiments, when the carrier wave Mixk is selected from the signals ϕ_(π/2) and ϕ_(−π/2) during a certain burst, the digital data stream q outputted from the AD converter 15 comprises information of a noise component of the sensing signal f_(k) during the certain burst, in place of the component of the sensing signal f_(k) effective for proximity sensing.

In one or more embodiments, the digital outputs of the decoder circuitry 16 are represented by an output tensor b_(X). In one or more embodiments, the output tensor b_(X) can be represented by the following expression (5):

$\begin{matrix} {{b_{X} = {XEa}},} & (5) \end{matrix}$

where X is a decode tensor which represents matrix operations performed on the digital data stream q by the decoder circuitry 16 during bursts #1 to #(2N).

In one or more embodiments, the decode tensor X can demodulate the input tensor a modulated with the encode tensor E when the product XE of the encode tensor E and the decode tensor X recited in expression (5) has only one non-zero element in each row.

In one or more embodiments, each of the carrier wave Mix1 to Mix(2N) is selected from the signals ϕ₀ and ϕ_(π) during two or more of bursts #1 to #(2N) to suppress detection of a motion artifact. In one or more embodiments, this operation increases the number of sampling of information of effective signal components for proximity sensing per each sensing frame with a reduced number of the AD converter 15, while effectively suppressing detection of a motion artifact. In one or more embodiments, information of effective signal components for proximity sensing are sampled a plurality of times in each sensing frame, and this allows increasing an effective sampling frequency, suppressing detection of a motion artifact.

In one or more embodiments, each of the carrier waves Mix1 to Mix(2N) is selected from the signals ϕ₀ and ϕ_(π) during two of more of the bursts #1 to #(2N), and accordingly there are two or more real elements in each column of the encode tensor E. In one or more embodiments, the product XE of the encode tensor E and the decode tensor X has one non-zero element in each row under such restriction, due to appropriate selection of the encode tensor E and the decode tensor X. In one or more embodiments, such selection of the encode tensor E and the decode tensor X is achieved through appropriate selection of the carrier waves Mix1 to Mix(2N) in the encoder circuitry 14 and appropriate design of the matrix operation performed in the decoder circuitry 16.

In one or more embodiments, the sampling timing of effective signal components for proximity sensing is evenly distributed in the time domain. In one or more embodiments, bursts in which the carrier wave Mixk, which is modulated with the sensing signal f_(k), is selected from the signals ϕ₀ and ϕ_(π) may be selected so that the selected bursts are not adjacent to each other in the time domain. In one or more embodiments, bursts in which the carrier wave Mixk is selected from the signals ϕ₀ and t and bursts in which the carrier wave Mixk is selected from the signals ϕ_(π/2) and ϕ_(−π/2) may be alternately disposed. This operation may effectively suppress detection of a motion artifact.

In one or more embodiments, sensing electrodes 3 from which information of effective signal components for proximity sensing may be spatially evenly distributed. In one or more embodiments, for a pair of carrier waves Mix(2p−1) and Mix(2p) modulated with sensing signals f_(2p-1) and f_(2p) associated with two sensing electrodes 3 spatially adjacent to each other, one of the carrier waves Mix(2p−1) and Mix(2p) is selected from the signals ϕ₀ and ϕ_(π) in a burst in which the other is selected from the signals ϕ_(π/2) and ϕ_(−π/2). This operation may effectively suppress detection of a motion artifact.

In one or more embodiments, the noise detection may be achieved based on mixer outputs obtained by modulating the signal ϕ_(π/2) or ϕ_(−π/2) with the sensing signals f₁ to f_(Q). In this case, the decoder circuitry 16 is designed to extract a noise component generated by modulating the signal ϕ_(π/2) or ϕ_(−π/2) with the sensing signals f₁ to f_(Q). In one or more embodiments, the timing at which information of noise components is obtained is evenly distributed in the time domain. In one or more embodiments, bursts in which the carrier wave Mixk, which is modulated with each sensing signal f_(k), is selected from the signals ϕ_(π/2) and ϕ_(−π/2) are selected from bursts #1 to #(2N) so that the selected bursts are not adjacent in the time domain. In one or more embodiments, the carrier wave Mixk is selected from the signals ϕ_(π/2) and ϕ_(−π/2) in every one burst. In one or more embodiments, the timing of obtaining the noise components may be spatially evenly distributed. In one or more embodiments, for a pair of carrier waves Mix(2p−1) and Mix(2p) modulated with sensing signals f_(2p-1) and f_(2p) associated with two sensing electrodes 3 spatially adjacent to each other, one of the carrier waves Mix(2p−1) and Mix(2p) is selected from the signals ϕ₀ and ϕ_(π) in a burst in which the other is selected from the signals ϕ_(π/2) and ϕ_(−π/2).

In one or more embodiments, the above-described conditions in which timing and positions for obtaining information of effective signal components and information of the noise components are evenly distributed can be described with an encode tensor E in which elements of real numbers and elements of pure imaginary numbers are arranged in a checkered pattern. In this case, the encode tensor E can be represented by the following expression (6a) with two encode matrices E₁ and E₂ represented by the following expressions (6b) and (6c), in one or more embodiments:

$\begin{matrix} {\mspace{79mu}{{E = {E_{1} + {iE}_{2}}},}} & \left( {6a} \right) \\ {{E_{1} = \begin{pmatrix} e_{11} & \; & 0 & \ldots & e_{1{({{2N} - 1})}} & \; & 0 \\ 0 & \; & e_{22} & \; & 0 & \; & e_{2{({2N})}} \\ \; & \vdots & \; & \ddots & \; & \vdots & \; \\ e_{{({{2N} - 1})}1} & \; & 0 & \ldots & e_{{({{2N} - 1})}{({{2N} - 1})}} & \; & 0 \\ 0 & \; & {e_{{({2N})}2}\;} & \; & 0 & \; & {{e\left( {2N} \right)}\left( {2N} \right)} \end{pmatrix}},\mspace{79mu}{and}} & \left( {6b} \right) \\ {\mspace{79mu}{{E_{2} = \begin{pmatrix} 0 & \; & e_{12} & \ldots & 0 & \; & e_{1{({2N})}} \\ e_{21} & \; & 0 & \; & e_{2{({{2N} - 1})}} & \; & 0 \\ \; & \vdots & \; & \ddots & \; & \vdots & \; \\ 0 & \; & e_{{({{2N} - 1})}2} & \; & 0 & \; & e_{{({{2N} - 1})}{({2N})}} \\ e_{{({2N})}1} & \; & 0 & \ldots & e_{{({2N})}{({{2N} - 1})}} & \; & 0 \end{pmatrix}},}} & \left( {6c} \right) \end{matrix}$

where the element e_(ij) of the encode matrix E₁ is a non-zero real number for i+j being an even number, and zero for i+j being an odd number, while the element e_(ij) of the encode matrix E₂ is zero for i+j being an even number, and a non-zero real number for i+j being an odd number.

In one or more embodiments, the encode matrices E₁ and E₂ can be represented by the following expressions (7a) and (7b) for a case where N is two, that is, the number of the sensing electrodes 3 for one AD converter 15 is four:

$\begin{matrix} {{E_{1} = \begin{pmatrix} e_{11} & 0 & e_{13} & 0 \\ 0 & e_{22} & 0 & e_{24} \\ e_{31} & 0 & e_{33} & 0 \\ 0 & e_{42} & 0 & e_{44} \end{pmatrix}},{and}} & \left( {7a} \right) \\ {E_{2} = {\begin{pmatrix} 0 & e_{12} & 0 & e_{14} \\ e_{21} & 0 & e_{23} & 0 \\ 0 & e_{32} & 0 & e_{34} \\ e_{41} & 0 & e_{43} & 0 \end{pmatrix}.}} & \left( {7b} \right) \end{matrix}$

In one or more embodiments, a 2×2 matrix O_(kl) in which one element is 1 and the remaining elements are 0 is defined by the following expression (8):

$\begin{matrix} {{O_{kl} = \left\{ {\delta_{ik}\delta_{jl}} \right\}},} & (8) \end{matrix}$

where δ_(ij) is the Kronecker delta. In one or more embodiments, the matrices O₁₁, O₂₂, O₁₂, and O₂₁ are represented by the following expressions (9a) to (9d):

$\begin{matrix} {{O_{11} = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}},} & \left( {9a} \right) \\ {{O_{22} = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}},} & \left( {9b} \right) \\ {{O_{12} = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}},{and}} & \left( {9c} \right) \\ {O_{21} = {\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}.}} & \left( {9d} \right) \end{matrix}$

In one or more embodiments, a partial matrix e_(kl) corresponding to the 2×2 matrix O_(kl) is defined for an encode matrix by the following expression (10):

$\begin{matrix} {{e_{kl} = \left\{ e_{{({k + {2{({i - 2})}}})}{({l + {({{2j} - 1})}})}} \right\}},} & (10) \end{matrix}$

where the partial matrix e_(kl) is an N×N matrix.

In one or more embodiments, the encode matrices E₁ and E₂ can be represented with the partial matrix e_(kl) and the matrix Oki by the following expressions (11a) and (11b):

$\begin{matrix} \begin{matrix} {E_{1} = {{\begin{pmatrix} e_{11} & \; & e_{13} & \; & e_{1{({{2N} - 1})}} \\ e_{31} & \; & e_{33} & \ldots & e_{3{({{2N} - 1})}} \\ \; & \vdots & \; & \; & \vdots \\ e_{{({{2N} - 1})}1} & \; & e_{{({{2N} - 1})}3} & \ldots & e_{{({{2N} - 1})}{({{2N} - 1})}} \end{pmatrix} \otimes \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}} +}} \\ {\begin{pmatrix} e_{22} & \; & e_{24} & \; & e_{2{({2N})}} \\ e_{42} & \; & e_{44} & \ldots & e_{4{({2N})}} \\ \; & \vdots & \; & \ddots & \vdots \\ e_{{({2N})}2} & \; & e_{{({2N})}4} & \ldots & e_{{({2N})}{({2N})}} \end{pmatrix} \otimes \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}} \\ {{= {{e_{11} \otimes O_{11}} + {e_{22} \otimes O_{22}}}},{and}} \end{matrix} & \left( {11a} \right) \\ \begin{matrix} {E_{2} = {{\begin{pmatrix} e_{12} & \; & e_{14} & \ldots & e_{1{({2N})}} \\ e_{32} & \; & e_{34} & \; & e_{3{({2N})}} \\ \; & \vdots & \; & \ddots & \vdots \\ e_{{({{2N} - 1})}2} & \; & e_{{({{2N} - 1})}4} & \ldots & e_{{({{2N} - 1})}{({2N})}} \end{pmatrix} \otimes \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}} +}} \\ {\begin{pmatrix} e_{21} & \; & e_{23} & \ldots & e_{2{({{2N} - 1})}} \\ e_{41} & \; & e_{43} & \; & e_{4{({{2N} - 1})}} \\ \; & \vdots & \; & \ddots & \vdots \\ e_{{({2N})}1} & \; & e_{{({2N})}3} & \ldots & e_{{({2N})}{({{2N} - 1})}} \end{pmatrix} \otimes \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}} \\ {{= {{e_{12} \otimes O_{12}} + {e_{21} \otimes O_{21}}}},} \end{matrix} & \left( {11b} \right) \end{matrix}$

where the operator ⊗ is the Kronecker product.

In one or more embodiments, the encode matrices E₁ and E₂ can be represented by the following expressions (12a) and (12b) for a case where N is two, that is, the number of the sensing electrodes 3 for one AD converter 15 is four:

$\begin{matrix} {{E_{1} = {{\begin{pmatrix} e_{11} & e_{13} \\ e_{31} & e_{33} \end{pmatrix} \otimes \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}} + {\begin{pmatrix} e_{22} & e_{24} \\ e_{42} & e_{44} \end{pmatrix} \otimes \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}}}},{and}} & \left( {12a} \right) \\ {E_{2} = {{\begin{pmatrix} e_{12} & e_{14} \\ e_{32} & e_{34} \end{pmatrix} \otimes \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}} + {\begin{pmatrix} e_{21} & e_{23} \\ e_{41} & e_{43} \end{pmatrix} \otimes {\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}.}}}} & \left( {12b} \right) \end{matrix}$

In one or more embodiments, the partial matrices e₁₁ and e₂₂ of the encode matrix E₁ and the partial matrices e₁₂ and e₂₁ of the encode matrix E₂ are 2×2 matrices, and it would be understood that expressions (11a) and (11b) hold for N being 2 when these partial matrices are defined by the following expressions (13a) to (13d):

$\begin{matrix} {{e_{1} = \begin{pmatrix} e_{11} & e_{13} \\ e_{31} & e_{33} \end{pmatrix}},} & \left( {13a} \right) \\ {{e_{22} = \begin{pmatrix} e_{22} & e_{24} \\ e_{42} & e_{44} \end{pmatrix}},} & \left( {13b} \right) \\ {{e_{12} = \begin{pmatrix} e_{12} & e_{14} \\ e_{32} & e_{34} \end{pmatrix}},{and}} & \left( {13c} \right) \\ {e_{21} = {\begin{pmatrix} e_{21} & e_{23} \\ e_{41} & e_{43} \end{pmatrix}.}} & \left( {13d} \right) \end{matrix}$

In one or more embodiments, the encode tensor E is represented by the following expression (14):

$\begin{matrix} {{E = {\sum\limits_{m}{V_{m} \otimes E_{m}}}},} & (14) \end{matrix}$

where the encode matrix E_(m) corresponds to a basic linear transformation V_(m).

In one or more embodiments the decode tensor X is represented by the following expression (15):

$\begin{matrix} {{X = {\sum\limits_{m}{W_{m} \otimes X_{m}}}},} & (15) \end{matrix}$

where the decode matrix X_(m) corresponds to a basic linear transformation W_(m).

In this case, the output tensor b_(x) is represented by the following expression (16), in one or more embodiments:

$\begin{matrix} \begin{matrix} {{b_{X} = {XEa}},} \\ {= {\left( {\sum\limits_{k = 0}^{M - 1}\;{\left( V_{k + 1} \right) \otimes \left( {{- {\sum\limits_{m = 1}^{k}\;{X_{M - {({k - m})}}E_{m}}}} + {\sum\limits_{m = {k + 1}}^{M}\;{X_{({m - k})}E_{m}}}} \right)}} \right)a}} \\ {= {{\left( {V_{1} \otimes \left( {{X_{1}E_{1}} + {X_{2}E_{2}}} \right)} \right)a} + {\left( {V_{2} \otimes \left( {{{- X_{2}}E_{1}} + {X_{1}E_{2}}} \right)} \right){a.}}}} \end{matrix} & (16) \end{matrix}$

In the final line of expression (16), M is selected as being 2. In this case, matrices V₁ and V₂ are selected as basic linear transformations for expression (14), and matrices W₁ and W₂ are selected as basic linear transformations for expression (15), where the matrices V₁ and V₂ are orthogonal to each other, and the matrices W₁ and W₂ are orthogonal to each other.

In one or more embodiments, the product XE of the encode tensor E and the decode tensor X is represented by the following expression (17):

$\begin{matrix} \begin{matrix} {{XE} = {\sum\limits_{k = 0}^{M - 1}\;{\left( V_{k + 1} \right) \otimes \left( {{- {\sum\limits_{m = 1}^{k}\;{X_{M - {({k - m})}}E_{m}}}} + {\sum\limits_{m = {k + 1}}^{M}\;{X_{({m - k})}E_{m}}}} \right)}}} \\ {= {{V_{1} \otimes \left( {{X_{1}E_{1}} + {X_{2}E_{2}}} \right)} + {V_{2} \otimes {\left( {{{- X_{2}}E_{1}} + {X_{1}E_{2}}} \right).}}}} \end{matrix} & (17) \end{matrix}$

In one or more embodiments, the decode tensor X can decode the input tensor a modulated with the encode tensor E when the product XE, which is represented by expression (17), has only one non-zero element in each row.

In one or more embodiments, the decode matrices X₁ and X₂ can be represented by the following expressions (18) and (19) with the matrix Oki:

$\begin{matrix} \begin{matrix} {X_{1} = \begin{pmatrix} \begin{matrix} x_{11} & x_{12} \end{matrix} & \ldots & x_{1{({2N})}} \\ \begin{matrix} x_{21} & x_{22} \end{matrix} & \; & x_{2{({2N})}} \\ \vdots & \ddots & \vdots \\ \begin{matrix} x_{{({2N})}1} & x_{{({2N})}2} \end{matrix} & \ldots & x_{{({2N})}{({2N})}} \end{pmatrix}} \\ {= {{\begin{pmatrix} \begin{matrix} x_{11} & x_{13} \end{matrix} & \ldots & x_{1{({{2N} - 1})}} \\ \begin{matrix} x_{31} & x_{33} \end{matrix} & \; & x_{3{({{2N} - 1})}} \\ \vdots & \ddots & \vdots \\ \begin{matrix} x_{{({{2N} - 1})}1} & x_{{({{2N} - 1})}3} \end{matrix} & \ldots & x_{{({{2N} - 1})}{({{2N} - 1})}} \end{pmatrix} \otimes \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}} +}} \\ {{\begin{pmatrix} \begin{matrix} x_{12} & x_{14} \end{matrix} & \ldots & x_{1{({2N})}} \\ \begin{matrix} x_{32} & x_{34} \end{matrix} & \; & x_{3{({2N})}} \\ \vdots & \ddots & \vdots \\ \begin{matrix} x_{{({{2N} - 1})}2} & x_{{({{2N} - 1})}4} \end{matrix} & \ldots & x_{{({{2N} - 1})}{({2N})}} \end{pmatrix} \otimes \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}} +} \\ {{\begin{pmatrix} \begin{matrix} x_{21} & x_{23} \end{matrix} & \ldots & x_{2{({{2N} - 1})}} \\ \begin{matrix} x_{41} & x_{43} \end{matrix} & \; & x_{4{({{2N} - 1})}} \\ \vdots & \ddots & \vdots \\ \begin{matrix} x_{{({2N})}1} & x_{{({2N})}3} \end{matrix} & \ldots & x_{{({2N})}{({{2N} - 1})}} \end{pmatrix} \otimes \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}} +} \\ {\begin{pmatrix} \begin{matrix} x_{22} & x_{24} \end{matrix} & \ldots & x_{2{({2N})}} \\ \begin{matrix} x_{42} & x_{44} \end{matrix} & \; & x_{4{({2N})}} \\ \vdots & \ddots & \vdots \\ \begin{matrix} x_{{({2N})}2} & x_{{({2N})}4} \end{matrix} & \ldots & x_{{({2N})}{({2N})}} \end{pmatrix} \otimes \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}} \\ {{= {{x_{11} \otimes O_{11}} + {x_{12} \otimes O_{12}} + {x_{21} \otimes O_{22}} + {x_{22} \otimes O_{22}}}},{and}} \end{matrix} & (18) \\ \begin{matrix} {X_{2} = \begin{pmatrix} \begin{matrix} y_{11} & y_{12} \end{matrix} & \ldots & y_{1{({2N})}} \\ \begin{matrix} y_{21} & y_{22} \end{matrix} & \; & y_{2{({2N})}} \\ \vdots & \ddots & \vdots \\ \begin{matrix} y_{{({2N})}1} & y_{{({2N})}2} \end{matrix} & \ldots & y_{{({2N})}{({2N})}} \end{pmatrix}} \\ {= {{\begin{pmatrix} \begin{matrix} y_{11} & y_{13} \end{matrix} & \ldots & y_{1{({{2N} - 1})}} \\ \begin{matrix} y_{31} & y_{33} \end{matrix} & \; & y_{3{({{2N} - 1})}} \\ \vdots & \ddots & \vdots \\ \begin{matrix} y_{{({{2N} - 1})}1} & y_{{({{2N} - 1})}3} \end{matrix} & \ldots & y_{{({{2N} - 1})}{({{2N} - 1})}} \end{pmatrix} \otimes \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}} +}} \\ {{\begin{pmatrix} \begin{matrix} y_{12} & y_{14} \end{matrix} & \ldots & y_{1{({2N})}} \\ \begin{matrix} y_{32} & y_{34} \end{matrix} & \; & y_{3{({2N})}} \\ \vdots & \ddots & \vdots \\ \begin{matrix} y_{{({{2N} - 1})}2} & y_{{({{2N} - 1})}4} \end{matrix} & \ldots & y_{{({{2N} - 1})}{({2N})}} \end{pmatrix} \otimes \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}} +} \\ {{\begin{pmatrix} \begin{matrix} y_{21} & y_{23} \end{matrix} & \ldots & y_{2{({{2N} - 1})}} \\ \begin{matrix} y_{41} & y_{43} \end{matrix} & \; & y_{4{({{2N} - 1})}} \\ \vdots & \ddots & \vdots \\ \begin{matrix} y_{{({2N})}1} & y_{{({2N})}3} \end{matrix} & \ldots & y_{{({2N})}{({{2N} - 1})}} \end{pmatrix} \otimes \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}} +} \\ {\begin{pmatrix} \begin{matrix} y_{22} & y_{24} \end{matrix} & \ldots & y_{2{({2N})}} \\ \begin{matrix} y_{42} & y_{44} \end{matrix} & \; & y_{4{({2N})}} \\ \vdots & \ddots & \vdots \\ \begin{matrix} y_{{({2N})}2} & y_{{({2N})}4} \end{matrix} & \ldots & y_{{({2N})}{({2N})}} \end{pmatrix} \otimes \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}} \\ {{= {{y_{11} \otimes O_{11}} + {y_{12} \otimes O_{12}} + {y_{21} \otimes O_{22}} + {y_{22} \otimes O_{22}}}},} \end{matrix} & (19) \end{matrix}$

where the matrices x₁₁, x₁₂, x₂₁, and x₂₂ in expressions (18) are partial matrices of the decode matrix X₁, and the matrices y₁₂, y₂₁, and y₂₂ in expressions (19) are partial matrices of the decode matrix X₂.

In one or more embodiments, due to the characteristics of the matrix Oki, matrices which are included in the product XE of the encode tensor E and the decode tensor X and correspond to the basic linear transformations V₁ and V₂ are represented by the following expressions (20) and (21):

$\begin{matrix} {\mspace{85mu}\begin{matrix} {{{X_{1}E_{1}} + {X_{2}E_{2}}} = {{\left( {x_{11}e_{11}} \right) \otimes O_{11}} + {\left( {x_{12}e_{22}} \right) \otimes O_{12}} +}} \\ {{\left( {x_{21}e_{11}} \right) \otimes O_{21}} + {\left( {x_{22}e_{22}} \right) \otimes O_{22}} +} \\ {{\left( {y_{12}e_{21}} \right) \otimes O_{11}} + {\left( {y_{11}e_{12}} \right) \otimes O_{12}} +} \\ {{\left( {y_{22}e_{21}} \right) \otimes O_{21}} + {\left( {y_{21}e_{12}} \right) \otimes O_{22}}} \\ {= {{\left( {{x_{11}e_{11}} + {y_{12}e_{21}}} \right) \otimes O_{11}} +}} \\ {{\left( {{x_{12}e_{22}} + {y_{11}e_{12}}} \right) \otimes O_{12}} +} \\ {{\left( {{x_{21}e_{11}} + {y_{22}e_{21}}} \right) \otimes O_{21}} +} \\ {{\left( {{x_{22}e_{22}} + {y_{21}e_{12}}} \right) \otimes O_{22}},{and}} \end{matrix}} & (20) \\ {{{{- X_{2}}E_{1}} + {X_{1}E_{2}}} = {{\left( {{{- y_{11}}e_{11}} + {x_{12}e_{21}}} \right) \otimes O_{11}} + {\left( {{{- y_{12}}e_{22}} + {x_{11}e_{12}}} \right) \otimes O_{12}} + {\left( {{{- y_{21}}e_{11}} + {x_{22}e_{21}}} \right) \otimes O_{21}} + {\left( {{{- y_{22}}e_{22}} + {x_{21}e_{12}}} \right) \otimes {O_{22}.}}}} & (21) \end{matrix}$

In one or more embodiments, the following eight equations to be solved are obtained based on the above-described matrices corresponding to the basic linear transformations V₁ and V₂ from the condition that the product XE of the encode tensor E and the decode tensor X has only one non-zero element in each row:

$\begin{matrix} {{{{x_{11}e_{11}} + {y_{12}e_{21}}} = {{c_{1}\delta_{1\; g}O_{11}} + {c_{2}\delta_{1\; g}^{\prime}O_{22}}}},{{{x_{12}e_{22}} + {y_{11}e_{12}}} = {{c_{3}\delta_{2g}O_{11}} + {c_{4}\delta_{2\; g}^{\prime}O_{22}}}},{{{{- y_{11}}e_{11}} + {x_{12}e_{21}}} = {{c_{5}\delta_{3g}O_{11}} + {c_{6}\delta_{3\; g}^{\prime}O_{22}}}},{{{{- y_{12}}e_{22}} + {x_{11}e_{12}}} = {{c_{7}\delta_{4g}O_{11}} + {c_{8}\delta_{4\; g}^{\prime}O_{22}}}},{{{x_{21}e_{11}} + {y_{22}e_{21}}} = {{c_{9}\delta_{1h}O_{11}} + {c_{10}\delta_{1\; h}^{\prime}O_{22}}}},{{{x_{22}e_{22}} + {y_{21}e_{12}}} = {{c_{11}\delta_{2h}O_{11}} + {c_{12}\delta_{2\; h}^{\prime}O_{22}}}},{{{{- y_{21}}e_{11}} + {x_{22}e_{21}}} = {{c_{13}\delta_{3h}O_{11}} + {c_{14}\delta_{3h}^{\prime}O_{22}}}},{{{and} - {y_{22}e_{22}} + {x_{21}e_{12}}} = {{c_{15}\delta_{4h}O_{11}} + {c_{16}\delta_{4h}^{\prime}O_{22}}}},} & (22) \end{matrix}$

where c₁ to c₁₆ are arbitrary constants, and g, g′, h, and h′ are any integer from one to four.

In one or more embodiments, when there is a solution to the eight equations (22), the product XE of the encode tensor E and the decode tensor X has only one non-zero element in each row, allowing demodulation of the input tensor a. In one embodiment, the partial matrices e₁₁, e₁₂, e₂₁, and e₂₂ of the encode matrices E₁ and E₂ are equal to an arbitrary regular matrix e, that is, it holds:

$\begin{matrix} {e_{11} = {e_{12} = {e_{21} = {e_{22} = {e.}}}}} & (23) \end{matrix}$

and the following expressions (24a) and (24b) hold for the partial matrices x₁₁, x₁₂, x₂₁, and x₂₂ of the decode matrix X₁ and the partial matrices y₁₂, y₂₁ and y₂₂ of the decode matrix X₂:

$\begin{matrix} {{x_{11} = {y_{12} = {x_{21} = {{- y_{22}} = {2(e)^{- 1}}}}}},{and}} & \left( {24a} \right) \\ {{- y_{11}} = {x_{12} = {y_{21} = {x_{22} = {2{(e)^{- 1}.}}}}}} & \left( {24b} \right) \end{matrix}$

In one or more embodiments, the input tensor a is reproduced as the output tensor b_(x) through selecting the partial matrices of the encode matrices E₁ and E₂ and the decode matrices X₁ and X₂ so as to satisfy the above-described expressions (23), (24a) and (24b). In one or more embodiments, this allows obtaining information of the signal levels A₁ to A_(2N) of the sensing signals f₁ to f_(2N) as the digital outputs of the decode circuitry 16.

In one or more embodiments, the solution space of the eight equations (22) is narrowed by adding constraint conditions. In one or more embodiments, the circuit size of the decoder circuitry 16 is additionally reduced by appropriately defining the constraint conditions.

In one or more embodiments, the decode matrices are selected so that two or more different basic linear transformations do not appear in each column, and thereby the number of linear transformation circuits simultaneously operate in the decoder circuitry 16 is reduced to one. This is advantageous in terms of the circuit size and the control logic size.

In one or more embodiments, the variable matrices related to the decode matrices X₁ and X₂ are degenerated by using matrices X_(1a) and X_(2a) as the decode matrices X₁ and X₂ so that only four matrices x₁₁, x₂₁, y₁₂, and y₂₂ are variable matrices, where the matrix Xia is defined so that the elements in the even columns are zero as recited in expression (25), and the matrix X_(2a) is defined so that the elements in the odd columns are zero as recited in expression (26):

$\begin{matrix} \begin{matrix} {X_{1a} = \begin{pmatrix} x_{11} & 0 & \ldots & x_{1{({{2N} - 1})}} & 0 \\ x_{21} & 0 & \; & x_{2{({{2N} - 1})}} & 0 \\ \vdots & \; & \ddots & \vdots & \; \\ x_{{({{2N} - 1})}1} & 0 & \ldots & x_{{({{2N} - 1})}{({{2N} - 1})}} & 0 \\ x_{{({2N})}1} & 0 & \; & x_{{{({2N})})}{({{2N} - 1})}} & 0 \end{pmatrix}} \\ {{= {{x_{11} \otimes O_{11}} + {x_{21} \otimes O_{21}}}},{and}} \end{matrix} & (25) \\ \begin{matrix} {X_{2a} = \begin{pmatrix} 0 & y_{12} & \ldots & 0 & y_{1{({2N})}} \\ 0 & y_{22} & \; & 0 & y_{2{({2N})}} \\ \; & \vdots & \ddots & \; & \vdots \\ 0 & y_{{({{2N} - 1})}2} & \ldots & 0 & y_{{({{2N} - 1})}{({2N})}} \\ 0 & y_{{({2N})}2} & \; & 0 & y_{{({2N})}{({2N})}} \end{pmatrix}} \\ {= {{y_{12} \otimes O_{12}} + {y_{22} \otimes {O_{22}.}}}} \end{matrix} & (26) \end{matrix}$

In this case, the matrices which are included in the product XE and correspond to the basic linear transformations V₁ and V₂ are represented by the following expressions (27) and (28), in one or more embodiments:

$\begin{matrix} {{{{X_{1a}E_{1}} + {X_{2a}E_{2}}} = {{\left( {{x_{11}e_{11}} + {y_{12}e_{21}}} \right) \otimes O_{11}} + {\left( {{x_{21}e_{11}} + {y_{22}e_{21}}} \right) \otimes O_{21}}}},{and}} & (27) \\ {{{{- X_{2a}}E_{1}} + {X_{1a}E_{2}}} = {{\left( {{{- y_{12}}e_{22}} + {x_{11}e_{12}}} \right) \otimes O_{12}} + {\left( {{{- y_{22}}e_{22}} + {x_{21}e_{12}}} \right) \otimes {O_{22}.}}}} & (28) \end{matrix}$

In one or more embodiments, the condition to reproduce the input tensor a, that is, the condition that the product XE of the encode tensor E and the decode tensor X has only one non-zero element in each row can be more restricted as recited in expressions (29):

$\begin{matrix} {{{{x_{11}e_{11}} + {y_{12}e_{21}}} = {c_{1}\delta_{1\; g}I}},{{{{- y_{12}}e_{22}} + {x_{11}e_{12}}} = {c_{2}\delta_{2\; g}I}},{{{x_{21}e_{11}} + {y_{22}e_{21}}} = {c_{3}\delta_{2\; g}I}},{{{and} - {y_{22}e_{22}} + {x_{21}e_{12}}} = {c_{4}\delta_{1\; g}I}},} & (29) \end{matrix}$

where I is the unit matrix, c₁, c₂, c₃, and c₄ are arbitrary constants, and g is an integer of 1 or 2. This is based on the fact that, if a plurality of non-zero elements appear in the same row of the matrices corresponding to the basic linear transformations V₁ and V₂, this is a valueless and redundant decode result.

There actually exists a solution to the four equations (29), although the solution is non-unique because of the existence of eight variable matrices and one variable g. For example, the partial matrices e₁₁, e₂₁, e₁₂, e₂₂, X₁₁, X₂₁, y₁₂, and y₂₂ satisfying the expressions (23) and (24a) are a solution to the four equations (29).

In one or more embodiments, the variable matrices related the decode matrices X₁ and X₂ are degenerated by using matrices X₁ b and X₂ b as the decode matrices X₁ and X₂ so that only four matrices x₁₂, x₂₂, y₁₁, and y₂₁ are variable matrices, where the matrix X₁ b is defined so that the elements in the odd columns are zero as recited in expression (30) and the matrix X_(2b) is defined so that the elements in the even columns are zero as recited in expression (31):

$\begin{matrix} \begin{matrix} {X_{1b} = \begin{pmatrix} 0 & x_{12} & \ldots & 0 & x_{1{({2N})}} \\ 0 & x_{22} & \; & 0 & x_{2{({2N})}} \\ \; & \vdots & \ddots & \; & \vdots \\ 0 & x_{{({{2N} - 1})}2} & \ldots & 0 & x_{{({{2N} - 1})}{({2N})}} \\ 0 & x_{{({2N})}2} & \; & 0 & x_{{({2N})}{({2N})}} \end{pmatrix}} \\ {{= {{x_{12} \otimes O_{12}} + {x_{22} \otimes O_{22}}}},{and}} \end{matrix} & (30) \\ \begin{matrix} {X_{2b} = \begin{pmatrix} y_{11} & 0 & \ldots & y_{1{({{2N} - 1})}} & 0 \\ y_{21} & 0 & \; & y_{2{({{2N} - 1})}} & 0 \\ \vdots & \; & \ddots & \vdots & \; \\ y_{{({{2N} - 1})}1} & 0 & \ldots & y_{{({{2N} - 1})}{({{2N} - 1})}} & 0 \\ y_{{({2N})}1} & 0 & \; & y_{{({2N})}{({{2N} - 1})}} & 0 \end{pmatrix}} \\ {= {{y_{11} \otimes O_{11}} + {y_{21} \otimes {O_{21}.}}}} \end{matrix} & (31) \end{matrix}$

In this case, the matrices which are included in the product XE and correspond to the basic linear transformations V₁ and V₂ are represented by the following expressions (32) and (33), in one or more embodiments:

$\begin{matrix} {{{{X_{1b}E_{1}} + {X_{2b}E_{2}}} = {{\left( {{x_{12}e_{22}} + {y_{11}e_{12}}} \right) \otimes O_{12}} + {\left( {{x_{22}e_{22}} + {y_{21}e_{12}}} \right) \otimes O_{22}}}},{and}} & (32) \\ {{{{- X_{2b}}E_{1}} + {X_{1b}E_{2}}} = {{\left( {{{- y_{11}}e_{11}} + {x_{12}e_{21}}} \right) \otimes O_{11}} + {\left( {{{- y_{21}}e_{11}} + {x_{22}e_{21}}} \right) \otimes {O_{21}.}}}} & (33) \end{matrix}$

In one or more embodiments, the condition to demodulate the input tensor a, that is, the condition that the product XE of the encode tensor E and the decode tensor X has only one non-zero element in each row can be more restricted as recited in expressions (34):

$\begin{matrix} {{{{{{- y_{11}}e_{11}} + {x_{12}e_{21}}} = {c_{5}\delta_{1\; g}I}},{{{x_{12}e_{22}} + {y_{11}e_{12}}} = {c_{6}\delta_{2\; g}I}},{{{{- y_{21}}e_{11}} + {x_{22}e_{21}}} = {c_{7}\delta_{2\; g}I}},{and}}{{{x_{22}e_{22}} + {y_{21}e_{12}}} = {c_{8}\delta_{1\; g}{I.}}}} & (34) \end{matrix}$

There actually exists a solution to the four equations (34), although the solution is non-unique due to the existence of eight variable matrices and one variable g. For example, the partial matrices e₁₁, e₂₁, e₂₂, x₁₂, x₂₂, y₁₁ and y₂₁ satisfying the expressions (23) and (24b) are a solution to the four equations (34).

In one or more embodiments, after a solution satisfying the expression (22), (29) or (34) is obtained, the encoder circuitry 14 is designed to select the carrier waves Mix1 to Mix(2N) based on the encode tensor E of the obtained solution, and the decoder circuitry 16 is designed to perform product-sum operations in accordance with the decode tensor X of the obtained solution.

The values q₁ to q_(2N) of the digital data stream q outputted from the AD converter 15, which are represented by expression (4), incorporate information of noise components as well as information of the effective signal components, where the effective signal components correspond to information generated by modulating the signals Co and t with the sensing signal f₁ to f_(2N), and the noise components correspond to information generated by modulating the signals ϕ_(π/2) and ϕ_(−π/2) with the sensing signal f₁ to f_(2N). In one or more embodiments, noise detection may be achieved by designing the decoder circuitry 16 to extract information generated by modulating the signals ϕ_(π/2) and ϕ_(−π/2) with the sensing signals f₁ to f_(2N).

To extract information generated by modulating carrier waves having phases different from the phases of the signals ϕ₀ and ϕ_(π), for example, the signals ϕ_(π/2) and ϕ_(−π/2), with the sensing signals f₁ to f_(2N) in addition to information generated by modulating the signals Co and t with the sensing signals f₁ to f_(2N), in one or more embodiments, decode matrices X_(a) and X_(b) obtained by the following expressions (35) from the decode matrices X_(1a) and X_(2a) and the decode matrices X_(1b) and X_(2b) may be used in the decoder circuitry 16, where the decode matrices X_(1a) and X_(2a) are a solution to equation (29) and the decode matrices X_(1b) and X_(2b) are a solution to equation (34):

$\begin{matrix} {{{X_{a} = {X_{1a} - {i\; X_{2a}}}},{and}}{X_{b} = {X_{1b} - {i\;{X_{2b}.}}}}} & (35) \end{matrix}$

In one or more embodiments, the decode matrix X_(a) is applied to the digital data stream q to obtain the output tensor b_(x) so that odd elements thereof correspond to information generated by modulating the signal ϕ₀ or ϕ_(π) with the sensing signals f₁ to f_(2N), that is, information of effective signal components, and even elements thereof correspond to information generated by modulating the signal ϕ_(π/2) or ϕ_(−π/2) with the sensing signals f₁ to f_(2N), that is, information of noise components. In one or more embodiments, the decode matrix X_(b) is applied to the digital data stream q to obtain the output tensor b_(x) so that odd elements thereof correspond to information of noise components, and even elements thereof correspond to information of effective signal components. In one or more embodiments, the decode matrices X_(a) and X_(b) are applied to the digital data stream q outputted from the AD converter 15 in the decoder circuitry 16 to obtain digital outputs corresponding to the effective signal components and digital outputs corresponding to the noise components.

In one or more embodiments, the processing unit 5 may be configured to achieve proximity sensing based on the digital outputs corresponding to the effective signal components and achieve noise detection based on the digital outputs corresponding to the noise components. In one or more embodiments, when detecting existence of noise based on the digital outputs corresponding to the noise components in a sensing frame, the processing unit 5 may discard data of the digital outputs corresponding to the effective signal components obtained for this sensing frame. In such an embodiment, proximity sensing of an object to the sensor array 1 may be achieved based on the digital outputs corresponding to the effective signal components obtained in a sensing frame for which no noise is detected.

The above-described techniques of this disclosure is applicable to a case where Q is an odd number of three or more, for example, a case where Q is 2N−1 for N being an integer of two or more. In this case, a similar calculation to that described above is performed with an assumption that the signal level A_(2N) of the sensing signal f_(2N) is zero during all of bursts #1 to #(2N), in one or more embodiments. In one or more embodiments, the calculation result with respect to the sensing signal f_(2N) is discarded in the decoder circuitry 16.

In one or more embodiments the capacitive sensing circuitry 4 is configured based on a solution for a case where Q=4 and N=2, that is, one AD converter 15 is prepared for four sensing electrodes 3.

When N=2, the encode tensor E is a 4×4 matrix and the encode matrices E₁ and E₂ are represented by expressions (7a) and (7b), in one or more embodiments. In one or more embodiments, the partial matrices e₁₁, e₁₂, e₂₁, and e₂₂ of the encode matrices E₁ and E₂, the partial matrices x₁₁, x₂₁, x₁₂, and x₂₂ of the decode matrix X₁, and the partial matrices y₂₁, y₁₂, and y₂₂ of the decode matrix X₂ are all 2×2 matrices.

In one or more embodiments, all of the elements of the encoder matrices E₁ and E₂ are selected from +1 or −1 to eliminate a need for providing a variable gain stage for the encoder circuitry 14. This may be advantageous in terms of the circuit size. In one or more embodiments, a similar approach may be applied to the decode matrices X₁ and X₂ and the decoder circuitry 16. In view of this, in one or more embodiments, the partial matrices e₁₁, e₁₂, e₂₁, and e₂₂ of the encode matrices E₁ and E₂, the partial matrices x₁₁, x₂₁, x₁₂, and x₂₂ of the decode matrix Xu, and the partial matrices y₁₁, y₂₁, y₁₂, and y₂₂ of the decode matrix X₂ are selected from 16 2×2 matrices defined by the following expressions (36):

$\begin{matrix} {{{\pm M_{0}} = {\pm \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}}},{{\pm M_{1}} = {\pm \begin{pmatrix} 1 & {- 1} \\ {- 1} & 1 \end{pmatrix}}},{{\pm M_{2}} = {\pm \begin{pmatrix} 1 & {- 1} \\ 1 & {- 1} \end{pmatrix}}},{{\pm M_{3}} = {\pm \begin{pmatrix} 1 & 1 \\ {- 1} & {- 1} \end{pmatrix}}},{{\pm M_{4}} = {\pm \begin{pmatrix} 1 & 1 \\ 1 & {- 1} \end{pmatrix}}},{{\pm M_{5}} = {\pm \begin{pmatrix} {- 1} & 1 \\ 1 & 1 \end{pmatrix}}},{{\pm M_{6}} = {\pm \begin{pmatrix} 1 & {- 1} \\ 1 & 1 \end{pmatrix}}},{{{and} \pm M_{7}} = {\pm {\begin{pmatrix} 1 & 1 \\ {- 1} & 1 \end{pmatrix}.}}}} & (36) \end{matrix}$

In one or more embodiments, the partial matrices of the encode matrices and the decode matrices are all defined as being regular to simplify the calculation; this allows handling the partial matrices of the decode matrices as integer multiples of the inverse matrices of the partial matrices of the encode matrices. In one or more embodiments, the solution space is accordingly restricted in accordance with the following expressions (37) and (38):

$\begin{matrix} {e_{kl} \in \left\{ \begin{matrix} {{\pm M_{4}},} & {{\pm M_{5}},} & {{\pm M_{6}},} & {\left. {\pm M_{7}} \right\},} \end{matrix} \right.} & (37) \\ {x_{kl} \in \left\{ {{\begin{matrix} {{\pm M_{4}},} & {{\pm M_{5}},} & {{\pm M_{6}},} & {\left. {\pm M_{7}} \right\},} \end{matrix}{and}y_{kl}} \in \left\{ \begin{matrix} {{\pm M_{4}},} & {{\pm M_{5}},} & {{\pm M_{6}},} & {\left. {\pm M_{7}} \right\}.} \end{matrix} \right.} \right.} & (38) \end{matrix}$

In one or more embodiments, the restriction of the solution space with respect to the partial matrices x_(kl) and y_(kl) of the decode matrices is based on the relations recited in expressions (39):

$\begin{matrix} {{{\pm M_{4}} = {{\pm 2}\left( M_{4} \right)^{- 1}}},{{\pm M_{5}} = {{\pm 2}\left( M_{5} \right)^{- 1}}},{{\pm M_{6}} = {{\pm 2}\left( M_{7} \right)^{- 1}}},{{{and} \pm M_{7}} = {{\pm 2}{\left( M_{6} \right)^{- 1}.}}}} & (39) \end{matrix}$

In one or more embodiments, for the case where the partial matrices of the encode matrices are regular, g=1, and c₁=c₂=c₃=c₄=4, one of the most simple examples of the partial matrices of the encode matrices and the decode matrices satisfying the following expressions (40) is represented by the following expressions (41):

$\begin{matrix} {{{{x_{11}e_{11}} + {y_{12}e_{21}}} = {c_{1}\delta_{1\; g}I}},{{{{- y_{12}}e_{22}} + {x_{11}e_{12}}} = {c_{2}\delta_{2\; g}I}},{{{x_{21}e_{11}} + {y_{22}e_{21}}} = {c_{3}\delta_{2g}I}},{{{{- y_{22}}e_{22}} + {x_{21}e_{12}}} = {c_{4}\delta_{1\; g}I}},} & (40) \\ {{{e_{22} = {e_{21} = {e_{12} = e_{11}}}},{x_{11} = {y_{12} = {x_{21} = \left( e_{11} \right)^{- 1}}}},{and}}{y_{22} = {- {\left( e_{11} \right)^{- 1}.}}}} & (41) \end{matrix}$

In one or more embodiments, the solution space is restricted in accordance with expressions (37) and (38). In such an embodiment, with respect to partial matrices of the encode matrices and partial matrices of the decode matrices, a sum or difference of two matrix products becomes an integer multiple of the unit matrix only when the two matrix products are each an integer multiple of the unit matrix. In one or more embodiments, expression (22) can be accordingly rewritten into the following expressions (42) for g being 1:

$\begin{matrix} {{{{{x_{11}e_{11}} + {y_{12}e_{21}}} = {c_{1}I}},{{{{- y_{12}}e_{22}} + {x_{11}e_{12}}} = 0},{{{x_{21}e_{11}} + {y_{22}e_{21}}} = 0},{{{{- y_{22}}e_{22}} + {x_{21}e_{12}}} = {c_{4}I}},{{{{- y_{11}}e_{11}} + {x_{12}e_{21}}} = {c_{5}I}},{{{x_{12}e_{22}} + {y_{11}e_{12}}} = 0},{{{{- y_{21}}e_{11}} + {x_{22}e_{21}}} = 0},{and}}{{{x_{22}e_{22}} + {y_{21}e_{12}}} = {c_{8}{I.}}}} & (42) \end{matrix}$

In one or more embodiments, the following expressions (43) can be obtained from expressions (42):

$\begin{matrix} {{{x_{11} = {2\left( e_{11} \right)^{- 1}}},{{- y_{11}} = {{2\left( e_{11} \right)^{- 1}} = x_{11}}},{x_{12} = {2\left( e_{21} \right)^{- 1}}},{y_{12} = {{2\left( e_{21} \right)^{- 1}} = x_{12}}},{x_{21} = {2\left( e_{12} \right)^{- 1}}},{y_{21} = {{2\left( e_{12} \right)^{- 1}} = x_{21}}},{x_{22} = {2\left( e_{22} \right)^{- 1}}},{y_{22} = {{2\left( e_{22} \right)^{- 1}} = x_{22}}},{c_{1} = 4},{c_{2} = 4},{c_{3} = 4},{and}}{c_{4} = 4.}} & (43) \end{matrix}$

In one or more embodiments, coefficients of 2 appear in expressions (43), because partial matrices of the decode matrices have coefficients of 2, when the partial matrices of the decode matrices are integer multiples of the partial matrices of the encode matrices.

In one or more embodiments, the following four equations (44) in which the right sides are zero matrices are equivalent to one another:

$\begin{matrix} {{{{{- y_{12}}e_{22}} + {x_{11}e_{12}}} = 0},{{{x_{21}e_{11}} + {y_{22}e_{21}}} = 0},{{{x_{12}e_{22}} + {y_{11}e_{12}}} = 0},{{{and} - {y_{21}e_{11}} + {x_{22}e_{21}}} = 0},} & (44) \end{matrix}$

because, when products of regular matrices are equal, the inverse matrices thereof are equal. In one or more embodiments, this equivalence is obtained from the following expressions (45) and (46):

$\begin{matrix} {{{{x_{11}e_{12}} = {y_{12}e_{22}}},{\left( {x_{11}e_{12}} \right)^{- 1} = \left( {y_{12}e_{22}} \right)^{- 1}}}{{{\left( e_{12} \right)^{- 1}\left( x_{11} \right)^{- 1}} = {\left( e_{22} \right)^{- 1}\left( y_{12} \right)^{- 1}}},{{x_{21}e_{11}} = {{- y_{22}}e_{21}}},}} & (45) \\ {{{{- y_{11}} = x_{11}},{x_{12} = y_{12}},{y_{21} = x_{21}},\;{and}}{x_{22} = {- {y_{22}.}}}} & (46) \end{matrix}$

In one or more embodiments, based on this equivalence, a solution is obtained by searching two pairs of regular 2×2 matrices for which the matrix products are equal to each other. In one or more embodiments, there is a plurality of pairs satisfying this requirement.

Because of the characteristics of a matrix product, a matrix product remains unchanged even when two columns of the multiplying matrix and two rows of the matrix to be multiplied are exchanged at the same time. Furthermore, a matrix product remains unchanged even when the sign of a column of the multiplying matrix and signs of corresponding two rows of the matrix to be multiplied are inverted at the same time. With respect to solutions related to partial matrices e_(kl) of the encode matrices to be multiplied, based on the relationships represented by the following expressions (47):

$\begin{matrix} {{{\pm M_{4}} = {{\pm \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}}M_{6}}},{{\pm M_{4}} = {{\pm \begin{pmatrix} 0 & {- 1} \\ 1 & 0 \end{pmatrix}}M_{5}}},{{{and} \pm M_{4}} = {{\pm \begin{pmatrix} 1 & 0 \\ 0 & {- 1} \end{pmatrix}}M_{7}}},} & (47) \end{matrix}$

if one of the simplest solution representable by the following expression (48) is obtained, remaining solutions are readily obtained, in one or more embodiments:

$\begin{matrix} {e_{kl} = {M_{4}.}} & (48) \end{matrix}$

For example, when a solution is selected to satisfy the following expressions (49), the sum of the row sums of two encode matrices always becomes two:

$\begin{matrix} {{{e_{11} = M_{4}},{e_{12} = M_{5}},{e_{21} = M_{6}},{e_{22} = M_{7}},{x_{11} = M_{4}},{y_{12} = M_{7}},{x_{21} = M_{5}},{{- y_{22}} = M_{6}},{{- y_{11}} = M_{4}},{x_{12} = M_{7}},{y_{21} = M_{5}},{and}}{x_{22} = {M_{6}.}}} & (49) \end{matrix}$

Expressions (49) represents one of solutions which allows demodulating the input tensor a.

As a detailed representation of the solution represented by expression (49), the encode tensor E and the decode matrices X_(a) and X_(b) can be represented by the following expressions (50) and (51), in one or more embodiments:

$\begin{matrix} {{E = {{E_{1} + {iE}_{2}} = {\begin{pmatrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & {- 1} & 0 \\ 0 & {- 1} & 0 & 1 \end{pmatrix} + {i\begin{pmatrix} 0 & {- 1} & 0 & 1 \\ 1 & 0 & {- 1} & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \end{pmatrix}}}}},{X_{a} = {{X_{1a} - {i\; X_{2a}}} = {\begin{pmatrix} 1 & 0 & 1 & 0 \\ {- 1} & 0 & 1 & 0 \\ 1 & 0 & {- 1} & 0 \\ 1 & 0 & 1 & 0 \end{pmatrix} - {i\begin{pmatrix} 0 & 1 & 0 & 1 \\ 0 & {- 1} & 0 & 1 \\ 0 & {- 1} & 0 & 1 \\ 0 & {- 1} & 0 & {- 1} \end{pmatrix}}}}},{and}} & (50) \\ {X_{b} = {{X_{1b} - {iX}_{2b}} = {\begin{pmatrix} 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & {- 1} \\ 0 & {- 1} & 0 & 1 \\ 0 & 1 & 0 & 1 \end{pmatrix} - {{i\begin{pmatrix} {- 1} & 0 & {- 1} & 0 \\ {- 1} & 0 & 1 & 0 \\ {- 1} & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 \end{pmatrix}}.}}}} & (51) \end{matrix}$

In one or more embodiments, the products of the decode matrices X_(a) and X_(b) and the encode tensor E can be represented by the following expression (52):

$\begin{matrix} {{{{X_{a}E} = {\begin{pmatrix} 4 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 4 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} + {i\begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 4 \end{pmatrix}}}},{and}}\text{}{{X_{b}E} = {\begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 4 \end{pmatrix} + {{i\begin{pmatrix} 4 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 4 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}}.}}}} & (52) \end{matrix}$

In one or more embodiments, as illustrated in FIG. 4, the encoder circuitry 14 is configured in accordance with expression (50), and the decoder circuitry 16 is configured in accordance with expression (51), where four sensing electrodes 3 are prepared for one AD converter 15, that is, Q=4, and the encoder circuitry 14 comprises four mixers 24 ₁ to 24 ₄ receiving carrier waves Mix1 to Mix4, respectively.

In one or more embodiments, the decoder circuitry 16 comprises a digital demodulator 25, coefficient supplying circuitry 26, decimator/filters 27 and 28, and a decoder 29. In one or more embodiments, the digital demodulator 25 is configured to demodulate the digital data stream q with an in-phase local carrier which is in phase with the drive signals and an out-of-phase local carrier having a phase shifted from that of the in-phase local carrier by π/2. In one or more embodiments, the digital demodulator 25 comprises digital mixers 31 and 32. In one or more embodiments, the digital mixer 31 is configured to multiply the digital data stream q by coefficients Co corresponding to the in-phase local carrier, and the digital mixer 32 is configured to multiply the digital data stream q by coefficients ϕ_(π/2) corresponding to the out-of-phase local carrier, which has a phase delayed from that of the in-phase local carrier by π/2. In one or more embodiments, the coefficient supplying circuitry 26 is configured to supply the coefficients ϕ₀ to the digital mixer 31 and the coefficients ϕ_(π/2) to the digital mixer 32, in synchronization with a reference clock signal CLK_(REF).

In one or more embodiments, the decoder 29 comprises product-sum operation circuitries 33 and 34. In one or more embodiments, the product-sum operation circuitry 33 is configured to perform a product-sum operation corresponding to the decode matrix X_(a), and the product-sum operation circuitry 34 is configured to perform a product-sum operation corresponding to the decode matrix X_(b).

In one or more embodiments, the decimator/filter 27 is configured to distribute the output of the digital mixer 31 to the product-sum operation circuitries 33 and 34, and the decimator/filter 28 is configured to distribute the output of the digital mixer 32 to the product-sum operation circuitries 33 and 34.

In one or more embodiments, Q=4, that is, N=2 and the driver circuitry 12 supplies drive signals to the sensing electrodes 3 ₁ to 3 ₄ so that the waveform of the drive signals remains unchanged over bursts #1 to #4.

In one or more embodiments, as illustrated in FIG. 5, the mixers 24 ₁ to 24 ₄ use the signals ϕ_(π), ϕ_(−π/2), ϕ_(π), and ϕ_(π/2) as the carrier waves Mix1 to Mix4, respectively, and modulate the carrier waves Mix1 to Mix4 with the sensing signals f₁ to f₄, respectively, during burst #1. In one or more embodiments, this operation corresponds to the first row of the encode matrices E₁ and E₂ in equation (50). In one or more embodiments, the value q₁ of the digital data stream q outputted from the AD converter 15 during burst #1 is given by the following expression (53):

q ₁=(A ₁ +A ₃)ϕ_(π)−(A ₂ −A ₄)ϕ_(π/2).  (53)

In one or more embodiments, the digital mixers 31 and 32 outputs demodulated data B1.1 and B1.2 obtained by multiplying q₁ by the coefficients ϕ_(π) and ϕ_(π/2), respectively. In one or more embodiments, the demodulated data B1.1 and B1.2 can be represented by the following expressions (54a) and (54b):

$\begin{matrix} {{{B\; 1.1} = {{\left( {A_{1} + A_{3}} \right)\Phi_{0}} - {\left( {A_{2} - A_{4}} \right)\Phi_{\prod{/2}}}}},{and}} & \left( {54a} \right) \\ {{B\; 1.2} = {{\left( {A_{1} + A_{3}} \right)\Phi_{\prod{/2}}} + {\left( {A_{2} - A_{4}} \right){\Phi_{0}.}}}} & \left( {54b} \right) \end{matrix}$

In one or more embodiments, the decimator/filter 27 outputs the demodulated data B1.1 received from the digital mixer 31 to the product-sum operation circuitry 33, and the decimator/filter 28 outputs the demodulated data B1.2 received from the digital mixer 32 to the product-sum operation circuitry 34.

In one or more embodiments, as illustrated in FIG. 6, the mixers 24 ₁ to 24 ₄ use the signals ϕ_(π/2), ϕ_(π), ϕ_(−π/2), and ϕ_(π) as the carrier waves Mix1 to Mix4, respectively, and modulate the carrier waves Mix1 to Mix4 with the sensing signals f₁ to f₄, respectively, during burst #2. In one or more embodiments, this operation corresponds to the second row of the encode matrices E₁ and E₂ in equation (50). In one or more embodiments, the value q₂ of the digital data stream q outputted from the AD converter 15 during burst #2 is given by the following expression (55):

q ₂=(A ₁ −A ₃)ϕ_(π/2)+(A ₂ +A ₄)ϕ_(π).  (55)

In one or more embodiments, the digital mixers 31 and 32 outputs demodulated data B2.1 and B2.2 obtained by multiplying q₂ by the coefficients ϕ_(π) and ϕ_(π/2), respectively. In one or more embodiments, the demodulated data B2.1 and B2.2 can be represented by the following expressions (56a) and (56b):

$\begin{matrix} {{{B\; 2.1} = {{\left( {A_{1} - A_{3}} \right)\Phi_{\prod{/2}}} + {\left( {A_{2} + A_{4}} \right)\Phi_{0}}}},{and}} & \left( {56a} \right) \\ {{B\; 2.2} = {{{- \left( {A_{1} - A_{3}} \right)}\Phi_{0}} + {\left( {A_{2} + A_{4}} \right){\Phi_{\prod{/2}}.}}}} & \left( {56b} \right) \end{matrix}$

In one or more embodiments, the decimator/filter 27 outputs the demodulated data B2.1 received from the digital mixer 31 to the product-sum operation circuitry 34, and the decimator/filter 28 outputs the demodulated data B2.2 received from the digital mixer 32 to the product-sum operation circuitry 33.

In one or more embodiments, as illustrated in FIG. 7, the mixers 24 ₁ to 24 ₄ use the signals ϕ_(π), ϕ_(π/2), ϕ_(π), and ϕ_(−π/2) as the carrier waves Mix1 to Mix4, respectively, and modulate the carrier waves Mix1 to Mix4 with the sensing signals f₁ to f₄, respectively, during burst #3. In one or more embodiments, this operation corresponds to the third row of the encode matrices E₁ and E₂ in equation (50). In one or more embodiments, the value q₃ of the digital data stream q outputted from the AD converter 15 during burst #3 is given by the following expression (57):

q ₃=(A ₁ −A ₃)ϕ_(π)+(A ₂ +A ₄)ϕ_(π/2).  (57)

In one or more embodiments, the digital mixers 31 and 32 output demodulated data B3.1 and B3.2 obtained by multiplying q₃ by the coefficients ϕ_(π) and ϕ_(π/2), respectively. In one or more embodiments, the demodulated data B3.1 and B3.2 can be represented by the following expressions (58a) and (58b):

$\begin{matrix} {{{B\; 3.1} = {{\left( {A_{1} - A_{3}} \right)\Phi_{0}} + {\left( {A_{2} + A_{4}} \right)\Phi_{\prod{/2}}}}},{and}} & \left( {58a} \right) \\ {{B\; 3.2} = {{\left( {A_{1} - A_{3}} \right)\Phi_{\prod{/2}}} - {\left( {A_{2} + A_{4}} \right){\Phi_{0}.}}}} & \left( {58b} \right) \end{matrix}$

In one or more embodiments, the decimator/filter 27 outputs the demodulated data B3.1 received from the digital mixer 31 to the product-sum operation circuitry 33, and the decimator/filter 28 outputs the demodulated data B3.2 received from the digital mixer 32 to the product-sum operation circuitry 34.

In one or more embodiments, as illustrated in FIG. 8, the mixers 24 ₁ to 24 ₄ use the signals ϕ_(π/2), ϕ_(π), ϕ_(π/2), and ϕ_(π) as the carrier waves Mix1 to Mix4, respectively, and modulate the carrier waves Mix1 to Mix4 with the sensing signals f₁ to f₄, respectively, during burst #4. In one or more embodiments, this operation corresponds to the fourth row of the encode matrices E₁ and E₂ in equation (50). In one or more embodiments, the value q₄ of the digital data stream q outputted from the AD converter 15 during burst #4 is given by the following expression (59):

q ₄=(A ₁ ±A ₃)ϕ_(π/2)−(A ₂ −A ₄)ϕ_(π).  (59)

In one or more embodiments, the digital mixers 31 and 32 outputs demodulated data B4.1 and B4.2 obtained by multiplying q₄ by the coefficients ϕ_(π) and ϕ_(π/2), respectively. In one or more embodiments, the demodulated data B4.1 and B4.2 can be represented by the following expressions (60a) and (60b):

B4.1=(A ₁ +A ₃)ϕ_(π/2)−(A ₂ −A ₄)ϕ_(π), and  (60a)

B4.2=−(A ₁ +A ₃)ϕ_(π)−(A ₂ −A ₄)ϕ_(π/2).  (60b)

In one or more embodiments, the decimator/filter 27 outputs the demodulated data B4.1 received from the digital mixer 31 to the product-sum operation circuitry 34, and the decimator/filter 28 outputs the demodulated data B4.2 received from the digital mixer 32 to the product-sum operation circuitry 33.

In one or more embodiments, the product-sum operation circuitries 33 and 34 of the decoder 29 performs operations corresponding to the decode matrices X_(a) and X_(b) on the data B1.1 to B4.1 and B1.2 to B4.2 received from the decimator/filters 27 and 28 during bursts #1 to #4 to calculate digital outputs 4A₁ϕ_(π), 4A₂ϕ₀, 4A₃ϕ₀, and 4A₄ϕ₀, which correspond to effective signal components, and digital outputs 4A₁ϕ_(π/2), 4A₂ϕ_(π/2), 4A₃ϕ_(π/2), and 4A₄ϕ_(π/2), which correspond to noise components. In one or more embodiments, the decode matrices X_(a) and X_(b) recites the unit imaginary number i, and accordingly the product-sum operation circuitries 33 and 34 are configured to use values outputted from the digital mixer 32 in performing multiplication by the unit imaginary number i; the product-sum operation circuitries 33 and 34 themselves are configured to perform operations in the real number domain, in one or more embodiments.

In one or more embodiments, the product-sum operation circuitry 33 calculates the digital outputs 4A₁ϕ₀, 4A₂ϕ_(π/2), 4A₃ϕ₀, and 4A₄ϕ_(π/2) by performing a matrix operation on the demodulated data B1.1, B2.2, B3.1, and B4.2 received during bursts #1 to #4, in accordance with the following expression (61):

$\begin{matrix} {\begin{pmatrix} {4A_{1}\Phi_{0}} \\ {4A_{2}\Phi_{\pi/2}} \\ {4A_{3}\Phi_{0}} \\ {4A_{4}\Phi_{\pi/2}} \end{pmatrix} = {\begin{pmatrix} 1 & {- 1} & 1 & {- 1} \\ {- 1} & 1 & 1 & {- 1} \\ 1 & 1 & {- 1} & {- 1} \\ 1 & 1 & 1 & 1 \end{pmatrix}{\begin{pmatrix} {B\; 1.1} \\ {{B2}{.2}} \\ {B\; 3.1} \\ {B\; 4.2} \end{pmatrix}.}}} & (61) \end{matrix}$

The 4×4 matrix of expression (61), which corresponds to the decode matrix X_(a), is represented in a form in which no unit imaginary numbers i are recited.

In one or more embodiments, the product-sum operation circuitry 34 calculates the digital outputs 4A₁ϕ_(π/2), 4A₂ϕ₀, 4A₃ϕ_(π/2), and 4A₄ϕ₀ by performing a matrix operation on the demodulated data B1.2, B2.1, B3.2, and B4.1 received during bursts #1 to #4, in accordance with the following expression (62):

$\begin{matrix} {\begin{pmatrix} {4A_{1}\Phi_{\pi/2}} \\ {4A_{2}\Phi_{0}} \\ {4A_{3}\Phi_{\pi/2}} \\ {4A_{4}\Phi_{0}} \end{pmatrix} = {\begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & {- 1} & {- 1} \\ 1 & {- 1} & {- 1} & 1 \\ {- 1} & 1 & {- 1} & 1 \end{pmatrix}{\begin{pmatrix} {B\; 1.2} \\ {{B2}{.1}} \\ {B\; 3.2} \\ {B\; 4.1} \end{pmatrix}.}}} & (62) \end{matrix}$

In one or more embodiments, the processing unit 5 is configured to perform arithmetic processing for proximity sensing of an object to the sensor array 1, based on the digital outputs 4A₁ϕ₀, 4A₂ϕ₀, 4A₃ϕ₀, and 4A₄ϕ₀, which correspond to the effective signal components.

In one or more embodiments, the processing unit 5 is further configured to perform noise detection based on the digital outputs 4A₁ϕ_(π/2), 4A₂ϕ_(π/2), 4A₃ϕ_(π/2), and 4A₄ϕ_(π/2), which correspond to the noise components. When the sensing signals f₁ to f₄ are free from an influence of noise, the digital outputs 4A₁ϕ_(π/2), 4A₂ϕ_(π/2), 4A₃ϕ_(π/2), and 4A₄ϕ_(π/2) are expected to be zero, in one or more embodiments. If not so, the sensing signals f₁ to f₄ are determined as experiencing an influence of noise.

Although various embodiments of this disclosure have been specifically described in the above, a person skilled in the art would appreciate that the technologies disclosed in this disclosure may be implemented with various modifications. For example, although embodiments in which self-capacitance sensing is performed have been described in the above, mutual capacitance sensing may be performed, in one or more embodiments. In this case, the sensor array 1 further comprises drive electrodes opposed to the sensing electrodes 3 ₁ to 3 _(Q), in one or more embodiments. In one or more embodiments, the driver circuitry 12 is configured to supply drive signals to the drive electrodes opposed to the sensing electrodes 3 ₁ to 3 _(Q).

Although FIG. 1 illustrates the arrangement in which sensing electrodes 3 associated with one AD converter 15 are arrayed in the vertical direction, the arrangement of sensing electrodes 3 associated with one AD converter 15 may be variously modified. As illustrated in FIG. 9, for example, when sensing electrodes 3 are grouped into Q groups 40 ₁ to 40 _(Q) arrayed in a specific direction, in the horizontal direction in FIG. 9, Q sensing electrodes 3 belonging to different groups 40 may be selected as sensing electrodes 3 associated with one AD converters 15. For example, the sensing electrodes 3 indicated by the hatchings in FIG. 9 may be selected as sensing electrodes 3 associated with one AD converters 15. 

What is claimed is:
 1. A system, comprising: sensing circuitry configured to: obtain sensing signals from sensing electrodes; generate first mixer output signals by modulating first carrier waves with the sensing signals; and combine the first mixer output signals; and processing circuitry configured to detect an object based on the combined first mixer output signals.
 2. The system of claim 1, wherein the sensing signals comprise: a first sensing signal associated with a first sensing electrode of the sensing electrodes, and wherein modulating the first carrier waves comprises modulating a first carrier wave of the first carrier waves with the first sensing signal during a first burst of a drive signal driven on the sensing electrodes to obtain the sensing signals.
 3. The system of claim 2, wherein the sensing signals further comprise a second sensing signal associated with a second sensing electrode of the sensing electrodes, and wherein modulating the first carrier waves further comprises modulating a second carrier wave of the first carrier waves with the second sensing signal during the first burst.
 4. The system of claim 3, wherein the sensing circuitry is further configured to generate second mixer output signals by: modulating a third carrier wave of second carrier waves with the first sensing signal during a second burst of the drive signal; and modulating a fourth carrier wave of the second carrier waves with the second sensing signal during the second burst.
 5. The system of claim 4, wherein the sensing circuitry is further configured to combine the second mixer output signals, and wherein the processing circuitry is further configured to detect noise based on the combined second mixer output signals.
 6. The system of claim 1, wherein the processing circuitry comprises: a digital demodulator configured to demodulate the combined first mixer output signals; and a decoder configured to generate a first digital output based on an output of the digital demodulator.
 7. The system of claim 6, wherein the processing circuitry further comprises: a decimator/filter configured to distribute the output of the digital demodulator to the decoder.
 8. The system of claim 6, wherein the decoder is configured to perform a product-sum operation based on a decode tensor to generate the first digital output.
 9. The system of claim 6, wherein the processing circuitry further comprises: coefficient supplying circuitry configured to supply a coefficient corresponding to an in-phase local carrier or an out-of-phase local carrier to the digital demodulator.
 10. A method for capacitive sensing comprising: obtaining sensing signals from sensing electrodes; generating first mixer output signals by modulating first carrier waves with the sensing signals; combining the first mixer output signals; and detecting an object based on the combined first mixer output signals.
 11. The method of claim 10, wherein the sensing signals comprise a first sensing signal associated with a first sensing electrode of the sensing electrodes and a second sensing signal associated with a second sensing electrode of the sensing electrodes; and wherein modulating the first carrier waves comprises: modulating a first carrier wave of the first carrier waves with the first sensing signal during a first burst of a drive signal driven on the sensing electrodes; and modulating a second carrier wave of the first carrier waves with the second sensing signal during the first burst.
 12. The method of claim 11, further comprising: generating second mixer output signals by: modulating a third carrier wave of second carrier waves with the first sensing signal during a second burst of the drive signal; and modulating a fourth carrier wave of the second carrier waves with the second sensing signal during the second burst.
 13. The method of claim 12, further comprising: combining the second mixer output signals; and detecting noise based on the combined second mixer output signals.
 14. A system, comprising: sensing circuitry configured to: obtain sensing signals from sensing electrodes; and generate an output signal based on the sensing signals; processing circuitry configured to detect an object based on the output signal from the sensing circuitry, wherein the processing circuitry comprises: an analog-to-digital converter configured to convert the output signal to a digital signal; a digital demodulator configured to demodulate the digital signal; and a decoder configured to perform a product-sum operation based on an output of the digital demodulator to generate a first digital output.
 15. The system of claim 14, wherein the sensing circuitry is further configured to: generate first mixer output signals by modulating first carrier waves with the sensing signals; and combine the first mixer output signals to generate the output signal.
 16. The system of claim 14, wherein the processing circuitry further comprises: a decimator/filter configured to distribute the output of the digital demodulator to the decoder.
 17. The system of claim 14, wherein the decoder is configured to perform the product-sum operation based on a decode tensor.
 18. The system of claim 14, wherein the processing circuitry further comprises: coefficient supplying circuitry configured to supply one or more coefficients each corresponding to an in-phase local carrier or an out-of-phase local carrier to the digital demodulator; and wherein the digital demodulator further comprises one or more mixers each configured to multiply the digital signal with a respective coefficient supplied by the coefficient supplying circuitry.
 19. The system of claim 14, wherein the decoder comprises one or more product-sum operation circuitries, wherein each of the one or more product-sum operation circuitries is configured to perform a product-sum operation corresponding to a decode matrix.
 20. The system of claim 14, wherein the processing system further comprises: a processing unit configured to perform arithmetic processing using the first digital output for proximity sensing of the object. 